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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">i</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c2">n</font> implies  <a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c2">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:20"/>
<font color="Maroon" title="c1">i</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c2">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<span class="kw">then </span><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="idea_1.html#K5" title="IDEA_1:func.5">NEG_N</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> = 
<font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c1">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D5" target="_self" onmouseover="rs('idea_1/D5')" onmouseout="rh()">Def5</a></span>
<br/>.= 
<a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c2">n</font></span>)</span>

;<br/>
<span class="kw">hence </span> <a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> (<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K6" title="IDEA_1:func.6">NEG_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c2">n</font>) = 
<span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T23" onmouseover="rs('nat_d/T23')" onmouseout="rh()">NAT_D:23</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T25" onmouseover="rs('nat_d/T25')" onmouseout="rh()">NAT_D:25</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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